For uniform sands & gravels a set of standard graphs is created for a spatial or transport volumetric concentration Cvs of 17.5% and a particle diameter of d=1 mm.

For other concentrations and particle diameters the graphs/curves may differ, especially the volumetric transport concentration Cvt graphs/curves.

The DHLLDV Framework is derived for uniform sands and gravels and constant spatial volumetric concentration.

The Hydraulic Gradient versus the Line Speed

The hydraulic gradient equals the mixture pressure loss Δpm divided by the liquid density ρl, the gravitational g constant and the length of the pipeline ΔL according to:

im=Δpm/(ρl·g·ΔL)

Multiplying the hydraulic gradient with a factor 1000 gives the mixture pressure loss for a ΔL=100 m pipe with water as the carrier liquid.

The Relative Excess Hydraulic Gradient versus the Line Speed

The Relative Excess Hydraulic Gradient Erhg is the solids effect (ρm-ρl) divided by the relative submerged density Rsd and the volumetric concentration either spatial Cvs or delivered Cvt.

Erhg=(ρm-ρl)/(Rsd·Cvs) or Erhg=(ρm-ρl)/(Rsd·Cvt)

The Relative Excess Hydraulic Gradient versus the Hydraulic Gradient

The Relative Excess Hydraulic Gradient Erhg is the solids effect (ρm-ρl) divided by the relative submerged density Rsd and the volumetric concentration either spatial Cvs or delivered Cvt.

Erhg=(ρm-ρl)/(Rsd·Cvs) or Erhg=(ρm-ρl)/(Rsd·Cvt)

The Hydraulic Gradient here is the carrier liquid pressure loss Δpl divided by the liquid density ρl, the gravitational g constant and the length of the pipeline ΔL according to:

il=Δpl/(ρl·g·ΔL)

The Head Losses on Durand & Condolios Coordinates

Durand & Condolios introduced a set of coordinates Ψ and Φ, based on using double logarithmic paper for processing experimental data into power curves. The ordinate and abscissa are given in the graph. Most of their experiments were carried out in pipes with a diameter of Dp=0.1524 m (6 inch). For larger diameter pipes the Durand & Condolios equation (grey solid line) over-estimates the pressure losses. For smaller pipe diameters it under-estimates the pressure losses.

The Bed Height/Fraction, the Slip Factor & the Spatial Concentration versus the Line Speed

The slip factor or holdup function determines the relation between the constant spatial volumetric concentration curves and the constant transport volumetric concentration curves.

Slip Factor=(1-ξ)=Cvt/Cvs and Slip Ratio=ξ=vsl/vls

The slip factor is determined based on the Limit Deposit Velocity. The bed height is determined based on the LDV and the slip factor.

The spatial volumetric concentration Cvs curve is based on constant delivered concentration Cvt and the slip ratio ξ curve.

The Influence Factors or Mobilization Factors versus the Line Speed

The mobilization factors show the collapse of the collisions in the heterogeneous flow regime and the mobilization of particles following the turbulent eddies in the homogeneous flow regime.

The Transition Heterogeneous-Homogeneous versus the Line Speed

The mobilized heterogeneous flow regime curve and mobilized homogeneous flow regime curve are added, resulting in a transition curve from the heterogeneous to the homogeneous flow regime.

The Construction of the Slip Ratio versus the Line Speed

The resulting slip ratio ξ curve is constructed based on 3 regions. The fixed or sliding bed region (the dark blue line for the fixed bed region and the light blue line for a sliding bed including sheet flow, 3LM), the region around the Limit Deposit Velocity (the red line) and the region of line speeds above the LDV (heterogeneous and homogeneous flow regimes). The solid green line give the resulting slip ratio.

The Actual Slip Velocity versus the Line Speed

The actual slip velocity is determined by multiplying the slip ratio curve with the line speed.